Problems For Partial Differential Equations Research Articles

Review and perspectives in quantum computing for partial differential equations in structural mechanics

Structural mechanics is commonly modeled by (systems of) partial differential equations (PDEs). Except for very simple cases where analytical solutions exist, the use of numerical methods is required to find approximate solutions. However, for many problems of practical interest, the computational cost of classical numerical solvers running on classical, that is, silicon-based computer hardware, becomes prohibitive. Quantum computing, though still in its infancy, holds the promise of enabling a new generation of algorithms that can execute the most cost-demanding parts of PDE solvers up to exponentially faster than classical methods, at least theoretically. Also, increasing research and availability of quantum computing hardware spurs the hope of scientists and engineers to start using quantum computers for solving PDE problems much faster than classically possible. This work reviews the contributions that deal with the application of quantum algorithms to solve PDEs in structural mechanics. The aim is not only to discuss the theoretical possibility and extent of advantage for a given PDE, boundary conditions and input/output to the solver, but also to examine the hardware requirements of the methods proposed in literature.

A fourth-order unfitted characteristic finite element method for free-boundary problems

A fourth-order unfitted characteristic finite element method (UCFEM) is proposed to solve free-boundary problems of time-dependent partial differential equations (PDEs). The boundary of the domain is implicitly driven by the solution of the PDE, while the PDE is proposed on the time-varying unknown domain. In this way, they form a coupled nonlinear system. The key ingredient of the UCFEM is to trace the domain explicitly with a fourth-order forward flow map and discretize the PDE in time with a fourth-order backward flow map. The PDE is solved with a fourth-order unfitted finite element method on a fixed mesh. Since the boundary of the domain is expressed explicitly with cubic spline functions, quadrature on cut elements can be done accurately and efficiently even the domain undergoes severe deformations. The UCFEM provides a framework of designing high-order numerical methods for free-boundary problems. We apply the UCFEM to a two-dimensional convection-diffusion equation and Navier-Stokes equations, and obtain the overall fourth order of accuracy. With extensive numerical experiments, we show that the proposed method can achieve the fourth-order convergence even on severely deformed domains.

Extending FEniCS to work in higher dimensions using tensor product finite elements

We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space–time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection–diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov–Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term In all cases, a p=1 basis is used and optimal L2 convergence rates of order hp+1 of the errors are achieved with respect to h refinement.

EXISTENCE FOR PARTIAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL CAUCHY-EULER OPERATOR

We investigate a boundary value problem for partial differential equation with Cauchy-Euler fractional operator $$\begin{aligned} x^{2\alpha }D_{*,x}^{2\alpha } u(x,y)+ x^{\alpha }D_{*,x}^{\alpha } u(x,y) + u_{yy}(x,y)=f(x,y),\ 0< x<\infty ,\ 0<y< 1,\ 0<\alpha < 1. \end{aligned}$$The limitations on convergence of series and the Mellin transform lead to the search for solutions in the class of functions, which possess continuous fractional derivatives for \(x\in (0,1)\) and equal to zero for \(x>1\). However, for some cases, the solution is found for the entire semi-infinite strip \(x>0\). The analysis is supported by the computations, which exhibit the high level of accuracy.

On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral

Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their properties are studied. In terms of the resolvent of the operator A, necessary and sufficient conditions are obtained for the existence of analytic resolving families of the equation under consideration. A perturbation theorem for such a class of operators is proved, and the Cauchy problem for the inhomogeneous equation with a distributed fractional derivative is studied. Abstract results are applied for the research of the unique solvability of initial boundary value problems for partial differential equations with a distributed derivative with respect to time.

On the examination of optical soliton pulses of Manakov system with auxiliary equation technique

Purpose:This manuscript addresses to investigate the optical soliton solution of the Manakov model which governs soliton transmission technology by utilizing the auxiliary equation technique. Methodology:After obtaining the nonlinear ordinary differential equation form (NODE) of investigated nonlinear partial differential equation problem (NLPDE) with the complex wave transform (cWT), the serial form is suggested in accordance with the utilized auxiliary method, and after written in the NODE form, the linear algebraic system is formed and by aid of the any symbolic calculation package the unknown parameters are obtained. In order to gain the soliton solutions, the proposed series form, the solution functions of the method, and the cWT are brought together and it is checked that the obtained solutions satisfy the main equation. By considering the functions that satisfy the main equation as solutions, graphical presentations are made and interpreted in order to better understand the obtained results. Findings:By applying the effective auxiliary equation method, optical soliton pulse solutions such as bright, singular, V-shaped, periodic singular, periodic bright, periodic bright–dark have been obtained, and it has been shown that the method can be applied effectively for such equations. Originality:The auxiliary equation method has not been applied to the Manakov model and the V-shaped, periodic singular, periodic bright soliton pulse solutions have not been reported before.

A Nonlinear Multigrid Method for the Parameter Identification Problem of Partial Differential Equations with Constraints

In this paper, we consider the parameter identification problem of partial differential equations with constraints. A nonlinear multigrid method is introduced to the process of parameter inversion. By keeping the objective functions on coarse grids consistent with those on fine grids, the proposed method reduces the dimensions of objective functions enormously and mitigates the risk of trapping in local minima effectively. Furthermore, constraints significantly improve the convergence ability of the method. We performed the numerical simulation based on the porosity identification of elastic wave equations in the fluid-saturated porous media, which suggests that the nonlinear multigrid method with constraints decreases the computational expenditure, suppresses the noise, and improves the inversion results.

A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations

Abstract Usually, to find the analytical and numerical solution of the boundary value problems of fractional partial differential equations is not an easy task; however, the researchers devoted their sincere attempt to find the solutions of various equations by using either analytical or numerical procedures. In this article, a very accurate and prominent method is developed to find the analytical solution of hyperbolic-telegraph equations with initial and boundary conditions within the Caputo operator, which has very simple calculations. This method is called a new technique of Adomian decomposition method. The obtained results are described by plots to confirm the accuracy of the suggested technique. Plots are drawn for both fractional and integer order solutions to confirm the accuracy and validity of the proposed method. Solutions are obtained at different fractional orders to discuss the useful dynamics of the targeted problems. Moreover, the suggested technique has provided the highest accuracy with a small number of calculations. The suggested technique gives results in the form of a series of solutions with easily computable and convergent components. The method is simple and straightforward and therefore preferred for the solutions of other problems with both initial and boundary conditions.

Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations

We introduce the concept of shape optimization of singular points of a weak solution in boundary value problems of partial differential equations (BVP) including fracture mechanics, shape optimization of boundary and interface, etc. Roughly speaking, singularity is a gap between weak and strong solutions. We also introduce the integral formula named the GJ-integral defined with an arbitrary subdomain as an extension of the J-integral in fracture mechanics. The GJ-integral is defined locally, is made from partial differential equations in BVP only, takes zero if the solution is regular, and expresses energy shape sensitivity. A generalization of the Hadamard variational formula is proposed and various cost functions are rewritten using the GJ-integral. In the last section there are numerical calculations using the GJ-integral with the finite element method.

A stochastic PDE approach to large N problems in quantum field theory: A survey

In this Review, we review some recent rigorous results on large N problems in quantum field theory, stochastic quantization, and singular stochastic partial differential equations (SPDEs) and their mean field limit problems. In particular, we discuss the O(N) linear sigma model on a two- and three-dimensional torus. The stochastic quantization procedure leads to a coupled system of N interacting Φ4 equations. In d = 2, we show uniformity in N bounds for the dynamics and convergence to a mean-field singular SPDE. For large enough mass or small enough coupling, the invariant measures [i.e., the O(N) linear sigma model] converge to the massive Gaussian free field, the unique invariant measure of the mean-field dynamics, in a Wasserstein distance. We also obtain tightness for certain O(N) invariant observables as random fields in suitable Besov spaces as N → ∞, along with exact descriptions of the limiting correlations. In d = 3, the estimates become more involved since the equation is more singular. We discuss in this case how to prove convergence to the massive Gaussian free field. The proofs of these results build on the recent progress of singular SPDE theory and combine many new techniques, such as uniformity in N estimates and dynamical mean field theory. These are based on joint papers with Scott Smith, Rongchan Zhu, and Xiangchan Zhu.

Distributed Adaptive Output Feedback Consensus of Parabolic PDE Agents on Undirected Networks.

In this article, we investigate the distributed adaptive consensus problem of parabolic partial differential equation (PDE) agents by output feedback on undirected communication networks, in which two cases of no leader and leader-follower with a leader are taken into account. For the leaderless case, a novel distributed adaptive protocol, namely, the vertex-based protocol, is designed to achieve consensus by taking advantage of the relative output information of itself and its neighbors for any given undirected connected communication graph. For the case of leader-follower, a distributed continuous adaptive controller is put forward to converge the tracking error to a bounded domain by using the Lyapunov function, graph theory, and PDE theory. Furthermore, a corollary that the tracking error tends to zero by replacing the continuous controller with the discontinuous controller is given. Finally, the relevant simulation results are further demonstrated to demonstrate the theoretical results obtained.

Boundary Value Problem for a First-Order Partial Differential Equation with Liouville Fractional Derivative

Boundary Value Problem for a First-Order Partial Differential Equation with Liouville Fractional Derivative

Existence and Uniqueness in the Linearised One and Two-dimensional Problem of Partial Differential Equations With Variational Method

The classical solution and the strong solution of a partial differential equation problem are continuously differentiable solutions. This solution has a derivative for a continuous infinity level. However, not all problems of partial differential equations can be easily obtained by strong solutions. Even the existence of a solution requires in-depth investigation. The variational formulation method can qualitatively analyze a single solution to a partial differential equation problem. This study provides an alternative method in analyzing the problem model of partial differential equations analytically. In this research, we will examine the partial differential equation modelling built from fluid dynamics modelling.

Local well-posedness for quasi-linear problems: A primer

Proving local well-posedness for quasi-linear problems in partial differential equations presents a number of difficulties, some of which are universal and others of which are more problem specific. On one hand, a common standard for what well-posedness should mean has existed for a long time, going back to Hadamard. On the other hand, in terms of getting there, there are by now both many variations—and also many misconceptions. The aim of these expository notes is to collect a number of both classical and more recent ideas in this direction, and to assemble them into a cohesive roadmap that can be then adapted to the reader’s problem of choice.

On an inverse problem for a nonlinear third order in time partial differential equation

In this article, first we convert an inverse problem of determining the unknown timewise terms of nonlinear third order in time partial differential equation (PDE) from knowledge of two boundary measurements to the auxiliary system of integral equations. Then, existence and uniqueness of the solution of this system is proved by means of the contraction principle on a small time interval. Also uniqueness of the solution of the inverse problem is given. However, since the governing equation is yet ill-posed (very slight errors in the temperature input may cause relatively significant errors in the output potential and source terms), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown terms. The third order in time PDE problem is discretized using the FDM and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both analytical and perturbed data are inverted. Numerical outcomes for two benchmark test examples are reported and discussed. The proposed numerical approach has also been discussed.

Predictor-feedback stabilization of two-input nonlinear systems with distinct and state-dependent input delays

This paper deals with the predictor-feedback stabilization problem of two-input nonlinear systems with distinct and state-dependent input delays. Since each individual channel may induce a different delay, a sequential approach is employed for the construction of two predictors such that the corresponding input delays can be compensated. Concretely, the predictor over the prediction horizon which corresponds to the smaller input delay is first constructed, and the predictor over the remaining time horizon that corresponds to the larger input delay is in turn constructed, by utilizing the first predictor signal as a basis. The feedback controllers are given based on the predictors. Due to the inherent constraint on the rates of the state-dependent input delays, regional stability results are achieved. Specifically, if the system is forward-complete, for the closed-loop nonlinear system, an estimation of the region of attraction is provided, and for the linear system, the exponential stability is guaranteed. The methodology is applied to the bilateral boundary control problem of hyperbolic partial differential equations (PDEs) with a moving boundary, for illustration of the effectiveness of the methodology.

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

A distributed optimal control problem for a semilinear parabolic partial differential equation is investigated. The stability of locally optimal solutions with respect to perturbations of the initial data is studied. Based on different types of sufficient optimality conditions for a local solution of the unperturbed problem, Lipschitz or Hölder stability with respect to perturbations are proved. Moreover, a particular example with semilinear equation, constant initial data, and standard quadratic tracking type objective functional is constructed that has at least two different locally optimal solutions. By the perturbation analysis, the existence of a problem with non-constant initial data is shown that also has at least two different locally optimal solutions.

On theoretical analysis of nonlinear fractional order partial Benney equations under nonsingular kernel

Abstract In the present article, the first step is devoted to develop some results about existence and uniqueness of solution to a general problem of fractional order partial differential equations (FPDEs) via classical fixed point theory. In the second step, a novel technique is used to handle the semi-analytical approximate solution for the considered general problem. Then, we extend the said result to fractional order partial Benney equations (FOPBEs) of the second and third order, which are special cases of the general problem we considered. We study the proposed problem under the Caputo-Febrizo fractional derivative (CFFD). With the help of the proposed method, we derive a series type approximate (semi-analytical) solution. Some numerical interpretations and visualizations are also given.

Chaos of the 2D linear hyperbolic equation with general van der Pol type boundary condition

This paper is concerned with the complex dynamics of the initial-boundary value problem of a 2D linear hyperbolic partial differential equation (PDE), where the parameter α that appeared in the general van der Pol type boundary condition is given by α∈R. The whole real line is divided into three intervals of the parameter to discuss the dynamics. The existence of chaos is first established in the sense of the exponential growth of total variation when the parameter locates in the central interval, which allows it to be positive, negative, or zero. By analyzing the chaotic dynamics of the piecewise continuous map induced by the hyperbolic PDE, such a PDE is further rigorously proved to be chaos in the interval of the positive parameter that is to the right of the central interval. Finally, the asymptotic behaviors of the hyperbolic PDE are systematically presented in the rest of the whole real line; more precisely, the hyperbolic PDE is either globally stable or unbounded.

Mathematical modeling of the gettering process for a cylindrical region

The paper considers a mathematical model of the technical process of external gettering of silicon rods. The purpose of this process is to remove unwanted impurities from silicon. The model of this process is an initial-boundary value problem for a linear partial differential equation of parabolic type. A feature of this problem is the presence of a nonlocal in time boundary condition, which reflects the law of conservation of mass (matter). For numerical modeling, an algorithm based on difference approximations of differential operators of the 1st and 2nd order is proposed. The classical sweep method is used to find a solution to a discrete problem.